CONTRIBUTIONS TO THE EINSTEIN-KURSUNOGLU FIELD EQUATIONS t By JOSEPH FRANCIS PIZZO JR. y A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1964 To Paula y ACKNOWLEDGMENTS The author extends his gratitude to the members of his committee. In particular, he is deeply grateful to Dr. J. Kronsbein, who suggested this problem and has given much of his time in helping to bring about the solution. The author would also like to thank both his wife, Paula for her help and suggestions in the preparation of the first draft, and Miss Nana Royer for the typing of the final copy. • • f I i I TABLE OF CONTENTS Page ACKNOWLEDGMENTS HI CHAPTER 1. INTRODUCTION 1 2. CURVATURE, DISPLACEMENT AND FIELD EQUATIONS .... k 3. REDUCTION AND SOLUTION 20 k. SPECIAL SOLUTIONS 30 Minkowski Space 30 A More General Case 33 5. SUMMARY 51 y BIBLIOGRAPHY 5h BIOGRAPHICAL SKETCH .... 56 Iv CHAPTER 1 INTRODUCTION In 1916, Einstein's classic paper, in which he formulated the general theory of relativity, appeared in Anna 1 en der Physik . [l] In this paper he was able to describe the phenomena of gravitation in terms of geometrical concepts. The field equations of gravitation were shown to be derivable from a variational principle, using a symmetric , second rank, covariant tensor. This tensor (which we will call the fundamental tensor , and will d^iote by g.,) represents the gravitational potentials. The theory was beautiful, and moreover It worked.' Einstein was still not satisfied. He reasoned that there are other fields in Nature besides gravitation. How do they fit into this picture? An example is the electromagnetic field and equations. There is no natural way for them to be included. To express the covariant electrodynamlc equations. Maxwell's equati'ons are written, then covariant derivatives are taken In place of ordinary partial derivatives. In other words, electrodynamics must be introduced separately. This type of inclusion has been con- sidered arbitrary and unsatisfactory by many theoreticians. Indeed, Einstein himself has stated, "A theory in which the gravitational field and the electromagnetic field do not enter as logically distinct 1. The numbers In square brackets refer to the bibliography at the end of the dissertation. I 2 structures would be much preferable." (See [7], page 115) This then is the aim of Einstein's unified field theory: to derive all fields from one, single non-symmetric tensor. Einstein formulated the unified field theory according to the same pattern he had used for his theory of gravitation, with the excep- tion that now he did not require the fundamental tensor to be symmetric. One of the resultant field equations turns out to be a set of sixty- four algebraic equations in sixty-four unknown functions of the g.. . These equations have to be solved before the other three field laws (second order partial differential equations) can be set up. This pro- blem occupied much of Einstein's later years, and although he found a solution to the sixty-four equations. It was In his own expressed opinion, too complicated to be of any further use. This is essentially where the problem has stood until this time. There has been no verification of Einstein's hypothesis, that other fields are included in the fundamental tensor, because the differential equations have never been solved. In this dissertation, we are able to show that, by a special restriction on the fundamental tensor, the unified field equations become those of gravitation and electrodynamics , while the components of g.. represent the gravitational and electro- I K magnetic fields. That which was introduced artificially before, now conies about naturally from one single tensor. In this special case. The geometrical concepts upon which the unified field theory is based, are developed In the second chapter. Also the different versions of the theory (Einstein's, Schroedinger 's , and Kursunoglu 's) are dis- cussed here. The correspondence of all of the theories to the gravita- tional equations of general relativity Is shown for limiting cases. J^F 3 Chapter three embodies our version of the theory. The sixty-four equations are shown to be reducible to twenty-four, which still cannot be solved in a useful form. Nevertheless, they suggest a modification in the fundamental tensor which allows a tractable form for the solution of the equations to be obtained. The three sets of differential equa- tions, referred to above, are constructed. One set is the same as the gravitational equations. The next is seen to be the first set of co- variant electrodynamic equations in the absence of charges and currents. The last is actually a set of third order partial differential equations and needs further investigation. It turns out we are able to solve this problem in our version of the theory. In chapter four two cases are considered. First, the equations are examined when the radius of the Einstein universe is taken infinite, in which case the gravitational equations are satisfied identically. The other two sets of equations are shown to be Maxwell's equations in the absence of matter. Next we take the more general case of a finite radius. The first two sets of differential equations were investigated before. Now, a terra by term examination reveals that the last field law Implies the second set of covariant electrodynamic equations. The last part of this chapter Is devoted to an exact solution of these equations in a special coordinate system. Throughout this dissertation an elementary understanding of tensor algebra and a basic knowledge of general relativity are assumed. CHAPTER 2 CURVATURE, DISPLACEMENT, AND FIELD EQUATIONS There are three varieties of unified field theory: Einstein's, Schroedinger 's, and Kursunoglu 's. Throughout each of these, two funda- mental entities are dominant; the displacement field and the curvature tensor. These concepts require some elaboration before they are used to derive the field equations. In relativity, we deal with quantities known as tensors, which have an appealing property; a tensor equation remains unchanged, re- gardless of the coordinate system in which it is expressed. That is, tensor equations are covariant. When the laws of physics are written as tensor equations, all reference frames are treated equally. The idea of a preferred system no longer exists. This formulation breaks down when we try to compare vectors at infinitesimal ly separated points. Consider, for example, the vector A at the point x and the vector « • • • A + dA at the point x + dx . The difference between the two is (a' + dA') - a' = dA' - (2.1) To illustrate the problem that has now arisen, suppose the vectors are 2 equal in a particular coordinate system. dA* = = a',j dx-". (2.2) 2. The coiwna indicates ordinary partial derivatives. 4 5 Requirements of covariance would demand that this relation be true in any coordinate system. Upon transformation -" A. {Th AZi dA' = -^ (A ) dx-" . (2.3) Since A is a vector, use can be made of its transformation law. dA' =-A(^a'^) ^'^'^' (2.4) dx Sx bK^ dA' = :^ dA*^ + a'^ -$^ dx'' . ^2-5^ ax^ ax'^ax^ The vanishing of the second term would insure the equality of the two vectors to be a covariant equation. Yet, in general, this is not so. An alternate way to require covariance of equation (2.2) is just to say that the difference between two vectors should transform like a vector. The difficulty in (2.5) is due to the fact that the vectors were compared at different points. It becomes necessary to find some pre- scription for translating vectors so they may be compared at the same point. This method is called parallel translation , and is accomplished in such a manner as to make the equality of vectors a covariant relation. That is, the difference between the vectors, when compared at the same point, will be a vector. The vector A translated parallel from x to x + dx wi 1 1 be denoted i i i i by A + 5A . Then at x + dx , the difference between the two vectors wi 1 1 be da' = (A* + dA*) - (a' + 5a') = dA* - 5a' . (2.6) 6 The expression 5a' will depend on the vector A and dx and can be represented by 6a' = -P'^^ a' dx* , (2.7) where the] is called the Displacement Field . Its components, which are to be determined, are functions of the x . In this form, da' = la'.^ + r'st ^'^ '^^^ • ^^'^^ The term in brackets is denoted by a special symbol, %t = ^''t^r'stA' ' (2-9) and is ca 1 1 ed a cova riant derivative . The difference, Da' , between the two vectors at the same point is to be a vector itself. This requires the covariant derivative to be a tensor. Therefore the transformation law for the displacement field must be P =^^^r +^ijsl_ (2.10) ' ^*"ax^ a^ ^1 ^^' ax" ^a^' It is clear that ordinary derivatives do not form a tensor and covariant derivatives must be taken Instead. Before the displacement field can be determined beyond its trans- formation law, (2.10), we must know more about the distribution of matter and charges which dictate the structure of the space. It Is clear that the displacement field is somehow related to the curvature of the space. So far, in this presentation, the notion of curvature has been a vague one, at best. Using the idea of parallel displacement, a mathematical picture of curvature may be displayed. Let a vector, A , be transferred parallel to itself along the boundary of an infinitesimal surface element and brought back to its starting f i point, X , «. We propose to compute the change, Aa , in the vector after one complete circuit. This change is given by Aa' = pA' (2.11) i Expression (2.7) is used for Sa'. The quantities j and A are ex- panded about their values at the Initial point, and terras kept to first order, since the curve is infinitely small. The expansions are s A^(xJ) (s) r pq A"- (s) (X^-X^(3)) (2.12) and r'st(^^''=r's. (s) r St,q J (x'' - x^/,J - (2.13) X ^5j (s; Let t t „t X -X ^3j = n The change in the vector is now written v^-r'^^^^A^fnw.r'strV^'^' (2.14) (2.15) With an appropriate renaming of dummy indices this becomes ^''[-r'pt.q^r's.r'pq^*'^'''"'- ('•''' It is understood that the terms preceding the integral are the initial values. As usual, we are interested in the displacement field and want to investigate the term in brackets to determine tensor character. It 8 can be shown that the Integral transforms as a tensor. Moreover, It is an antisymmetric tensor since ^n'^cn^ = f d(TiV) 'f-n^dr^ , (2.17) and the first term on the right hand side vanishes, because it is an 3 exact differential. The Integral will be denoted by f\^ » ^TiW . (2.18) In equation (2.16), the quantity In parentheses can be expressed as a sum of parts v^lch are symmetric and antisymmetric on the indices t and q. The contraction of the symmetric part with the Integral vanishes leaving ^' = .(p ^ -p -r T +r r jAPfir(2.i9) M pt,q I pq.t I stl pq > sql pV ^ ■^' It is knoMi that Aa Is a vector since It is obtained by parallel dis- placement. The term A f <• is a third rank tensor, therefore the tensor character of '^ ptq ~ "J pt.q ^ 1 pq.t * I stl pq " 1 sq I pt V • ) Is established. This is the wel 1 -known Riemann-Christof fel tensor. R Is also referred to as the curvature tensor, because whenever it ptq vanishes, there is no change in a vector after it Is displaced parallel to itself around a complete circuit. The space Is then said to have no curvature, or to be flat. Contraction on the indices i and q gives the second rank tensor. 3. The hook under the Indices Is used to indicate antisymmetry. 9 P q p pq rP pP pPpH ^PI'M .,+.,+ . , - I ., , (2.21) ik,p I ip,k iq pk qpl ik ^ ' which Is named the RfccI tensor, or the contracted curvature tensor. It is to play a centraJ role in the variational principle from which the field equations are to be derived. For a variational method, we postulate an invariant space integral which involves the displacement field. Invariance of this integral pre- supposes the existence of a scalar density, L, which can be formed by the contraction of the Ricci tensor with a contravariant tensor density, jj . The field laws are to be derived from 6/LdT = , (2.22) where dT is an element of four-volunie and L ' f" R.,^ (2.23) ik n^ is considered as a function of the H\ and | ., , which are to be varied independently, their variations vanishing on the boundary of integration. This postulate would lead to the equations of the purely gravi- tational field if the condition of symmetry were imposed on ^\ and pb ... (This will be demonstrated in the next chapter,) However, the present theory is an attanpt to generalize these equations, and the constructions which were essential for the setting up of covarJant 4. A scalar density transforms like a scalar but with the inverse of the Jacobian determinant included. This is necessary to cancel the Jacoblan which results from the transformation of the element of four- volume. The result Is that the integral transforms like an invariant. equations are independent of the assmnption of symmetry .' Instead of symmetry, an analogous condition is posited. It is referred to as Transposition Invariance. Even though ^j and I .. are non-sytranetric, their transformation laws would be invariant if the indices of these variables were to be transposed and then the free indices interchanged. This Is what is meant by transposition invariance. In addition to the variational principle, it will be postulated that all field laws shall be transposition invariant. If we proceed to substitute (2.21) for R., into equation (2.23) I K and carry out the variation indicated in (2.22), we would find that the resulting equations are not transposition invariant. To circumvent such a distasteful result, four new arbitrary variables, K, are intro- duced by making a formal change in the description of the field. In b place of p ., , the following substitution is made h * h *b b The I ., is a displacement field just as the ] ... The K are treated in the variational procedure as independent variables. After the 5. An example of an equation which is transposition invariant is s s Sik.b "^skf ib -^isF bk=° TranspKJSing the indices on the variables yields s s 9ki,b ■ ^ksP bi " 9sff i<b = ° • Interchanging the free indices (i and k) gives back the original equation .s „s 9ik,b ■ ^isF bk "• ^skP ib= ° It variation is carried out, the X^ will be chosen to make the field equations transposition invariant, and then they will be eliminated from the system. b The contracted curvature tensor, with P replaced by (2.24) f^ikT) = Rik^r) M^.^k-^ .) . (2.21a) * \^ere R.j^(p)is the same as expression (2.21), except that a star is b put over all the ["* .^. The variational integral is now /<5(^j''[R.,(r) M^j^k-\^.)]} dT=0 . (2.25) ik *^ After the variations in "^j ' , p .^, and X^ are made, the integrand, denoted by I, takes on the following appearance * :\, • » I * k . * i The last term is a generalized divergence and since it appears in a four- volume integral, it can be converted to an integral over the three- manifold enclosing this volume. The variations vanish on the boundary, hence the last term in equation (2.26) can be ignored. The variations in the remaining terms are all independent of each other so each coefficient must vanish separately, giving the three equations * ^•k^r) ^ (^i.k -\,o = ° (2-27) :> ij .k 12 k = (2.28) The four extra variables, \ , may be given any value. The three equa- tions will be transposition invariant '^ ^^ is chosen such that P .. = . (2.30) ik V Now V can be eliminated entirely from the equations by writing (2.27) as two equations. Rjj,= (2.31) R., . + R, . . + R. . , = (2.32) ik,b kb,i bi,k ^ ' V V V b The star may just as well be excluded from above the P ., , renaming them. By contraction on the indices k and s in (2.29), we can verify that the expression in parentheses vanishes and the equation can be written Before further simplification is made, it should be noted that the field law, (2.28) is already implied in this expression. This can be seen from the following consideration. The equation formed by exchanging free indices, k .. ? ■ . t jj .s-^^i) Pts-i! Pst-]) rst = °' (2.33a) 6. The dash under the indices indicates symmetry. is Just as valid, if both equations are contracted on the indices k and s and (2.33a) subtracted from (2.33), then clearly •s ^.si '11 .s-l^'.s)-» = 'll^3• (^-j-.) The covariant form of a tensor density is defined so that DipF^^^^llpir^ (2.35) Equation (2.33) is multiplied through by Ij.^ and the summation carried out to give r St = 2I1 ikir\s ' (2.36) which is replaced in (2.33) t.s-h]%^■f.n)"'^\s*])"\'\^'o. ,..3;, The definition of a density, ^flik 1/2 ik , ^ , 1/2 Fk IJ = 9 g = (-det. gj,^)'^ g"" . (2.38) and the rule for the derivative of a determinant, ^s = 99^ 9ab.s = - 99abg^^s ' (2-39) are used to bring (2.36) into the final form, s s ^ik.b - ^skP ib "9.3P ^^ = . (2.40) This set of sixty-four equations gives the relations for the sixty-four p .^ in terras of the g and their partial derivatives. Once the P I Ik 14 are known, they are substituted in the other three field equations, r ik=° (2-30) V R-i u + R. u • + Ru- I = = Rr., .■, , (2.32) ik,b kb, I bi ,k [ik,bj * \ j i V V V v for a solution of g.^^. This is the formalism of Einstein's unified field theory [5]. Schroedinger 's unified field theory is quite similar to that of Einstein's. Parallel transfer, displacement fields, and the curvature tensor all form the basis for the theory. The difference appears in the integrand of the variational principle. Whereas Einstein took ik ■U Rji^ as the .scalar density, Schroedinger chooses [I8] L=^(-det. R.^)^/2 . (2.41) which is the simplest scalar density that can be built out of the curvature tensor. The constants 2 and ^- have no influence on the result. A contravariant tensor density' is introduced here also: but it can be eliminated in the end, leaving equations which contain b only the P ... When the variation, 6/LdT = /-^ SR.j^dT = , (2.43) ik 7. The square brackets in (2.32) indicate summation over the cyclic permutations of the enclosed indices. This convention will be used hereafter. 15 is carried out with (2.41) as integrand, the result is the following set of equations, 9ik.b-9skP'ib-9isr'bk=0 (2.Z,i|) R., = Xg.^. (2.45) These can be taken as the field laws as they stand, but Schroedinger takes the theory further. Equation (2.45) can be substituted in (2.44) to eliminate the g., . The field equations are now sixty-four differential '" b equations involving nothing but the sixty-four P ., . R ik.b-f^skP ib-'^isThk^O • (2.46) This is known as the "purely affine theory." Kursunoglu's approach to the theory [16] retains the basic ideas of displacement field, curvature tensor, and the fundamental tensor ik g.j^ (or g ) but the scalar density for the variational integral is quite different and entails some new concepts. These will be defined before the variational principle and the field equations are displayed. The fundamental tensor, g., , is expressed in Kursunoglu's theory, I K in terms of its synmetric part, a,. , and its antisymmetric part, 4>., . 9ik=^k-^<2G)'/V\*., . (2.47) where G is the gravitational constant and r is a "fundamental length." o The determinant, g, of g., is constructed in terms of what will be called the "Kursunoglu invariants," namely SI = Y*'''*ik (2.48) 16 and ^ ] , v-1/2 ijkb^ ^ /~ i„\ X=3 (-a) eJ <»>..(t.^^ . (2.49) g = a(I + 2Gc ^r A - 4g c °r V) <2.50) o o The action principle from wliich the equations are to be deduced is *•^Ml'''^•|<■" 2'-o'^'^-b -^-a)]dT= . (2.51) where b is the determinant of the tensor, b., = a'''^g"'/^(a., + 2Gc'\ h. *^ ) . (2.52) ik ^^ik oisk' \ -^ J The variations carried out in (2,51) produces the Kursunoglu field equations. 9ik,b -SskP ib -9isr\k = ° ^2.53) '^ik=-'-o"'(^ik-^k^ (2.54) ■^[i^k.bl = V'(*ik.b ^ *kb.i ^ *bi.k) (2.55) The nomenclature, "fundamental length," is rationalized by examining the field equations. The vanishing of r gives the general relativis- tic case in the absence of charges. This existence of free charges is now associated with a finite fundamental length. All three field theories discussed here have at least one thing in common: they have been constructed so that there exists a correspond- ence principle which takes the unified field equations over the well- known gravitational equations of general relativity when the anti- symmetric field is absent. This is easy to see. First consider Einstein's V 17 set of equations and require the g,, to be composed of only a symmetric part, a.^. g., = a = a . (2.57) IK 1 k ki b The solution of equation (2.40) for the P is readily found by per- muting the indices to get three eojuations s s a., . - a , P .. - a. P . , = (2.40) ik,b sk I lb IS 1 bk ^ ' -■- ■ -^-sr ki -^ksT ib=° (2.58) s s 'kb,i °sbl ki " ^ksl ib s % au- ^ - a .P ,, - a. P , . = . (2.59) bi ,k SI 1 bk bs 1 ki ^ A combination of these three equations gives 5 Si u • + Su! I ~ a-i u = 2a . P , . . (2.60) kb, I bi ,k I k,b sb I ki ^ ' Equation (2.35) applies here b P ,. = ~a''P(a. . + a . , - a., ) (2.61) I ki 2 ^ kp,i pi,k ik,p' b The symmetry of the P . . is a consequence of the synmetry of the a. . The last notation is a matter of convention. Equation (2.61) is the "Christoffel symbol," where it is understood that only the symmetric part of the q., is used. The synmetric tensor, a , , is identified as the ^ik ik metric tensor of the space. Since equation (2.30) is satisfied identically, the X., may as well be chosen zero, in which case (2.31) and (2.32) could be recombined as 18 This is precisely the gravitational field equation in empty space. (See, for example, [9], page 81) A particular solution to these equa- tions corresponds to the field of an isolated particle continually at rest. The famous explanation of the discrepancy in the advance of Mercury's perihelion is a result of the solution to (2,63), There is no question about the physical significance of the gravitational field equations, so this gives a certain measure of confidence to the generali- zation. (2.30), (2.31), (2.32), and (2.40). A similar situation exists with Schroedinger 's theory in the limit of a symmetric g. . The field equations, (2.45), become the same as Einstein's with the addition of a term involving X, which is now identi- fied with the cosmological constant. Actually, the limiting case of Schroedinger 's theory is the original form of the gravitational equa- tions. The cosmological constant is so small that it need not be in- cluded on a scale such as our solar system, (See [9], p. 100) In Kursunoglu's theory, the correspondence to general relativity is achieved by the vanishing of the fundamental length. Again we revert to the equations of general relativity. This method is much like quantum mechanical correspondence to classical mechanics when Planck's constant vanishes. Therefore all three theories have at least some basis, due to the fact that they reduce in the limit to the well-known and tested equations of gravitation derived by Einstein in his general theory of relativity. What has been presented so far is only the problem: a non-symmetric, fundamental tensor, g. , is chosen to represent the fields in Nature. "^ b The displacement field, H .^, is introduced to insure that the field laws will not be dependent upon the choice of a coordinate system. From 19 a postulatory basis, a variational principle is used to derive a set of field equations. A solution to these equations will presumably give a description of nature in which all fields are united in the single tensor, g.j^, in the same manner as the electric and magnetic fields are unified in the electromagnetic tensor. Now that the problem is laid before us, the next step is to pro- vide a solution to these field equations. CHAPTER 3 REDUCTION AND SOLUTION It IS evident that the three versions of unified field theory, which have been presented, are little more than postulates. Their tenuous claim to validity comes from the correspondence to the known equations of general relativity and the correct count of functions. So far there has been no indication that they give a field description of Nature in which all fields are united in a single tensor. In keeping with this spirit, we choose to deal with yet another form of the equations, which can be considered as an adaptation of either Schroedinger 's or Kursunoglu's equations to an Einstein model of the universe. s s ^ik.b-^skP ib -SisP bk=° (3.1) « 2 9n, , i ?^ 4 ?^ k ik=) '^ - (3.2) , otherwise R., = (Je ^'^ Rr. 2 [ik,b] - ~ g2 9[lk,b] (3-3) k lik'O (3.^) V In the limit of a vanishing antisymnetric field, the antisynsnetric part of I .^ vanishes and the symmetric component, P .^, is the Christoffel symbol. Then (3.2) is seen to be exactly the set of differential equa- tions which describe the Einstein Cylindrical Model of the universe (See 20 [9], p. 159). Equations (3-3) and (3-4) are satisfied identically. Due to this correspondence, these equations should be considered as valid as the other three sets. In all four cases, it has not been possible to ascertain if the theory contains any information other than the simple reduction to the gravitational equations. The reason for this is that a simple, general b b expression for the P ,. has not been found. True, the P .. have been expressed in special coordinates, but the differential equations have not been solved using these values. In this chapter and the subsequent b ones an appeal ingly simple form for the P .. in a special case, along with new information from the field equations, will be presented. b The relation for P ., , (3-1) is correnon to all four versions. It I Ik ^ ' b is a set of sixty-four equations from which the sixty-four 1 .. are to be determined as functions of the sixteen g., and their derivatives. ik A solution to these equations has been given for a system of spherically syrranetric coordinates. [17, 21] Kursunoglu [16] sets up b the differential equations, (2.31) and (2,32), using the P .. expressed in terms of the g,. , but can offer no explicit solution. In fact, no solution has yet been given to equation (2.31) in these special coordi- nates. It Is for this reason that a completely general (coordinate inde- pendent) solution to (3.1) is desired. A little manipulation of (3.1) is enough to show that a general solution is far from trivial. Nevertheless, several formal istic solu- tions have been offered. [4, 10, 22] Formal istic, in this context, is b meant to Imply that an explicit expression for P ., is never written I Ik jj down. What is given is a prescription for determining P . . A brief danonstration of Einstein's formal istic solution [k] may illustrate this 22 s better. In (3.1), ggi^P -i. and g, P ,, are considered as individual terms and named 9skrib = ^ik|b ' (3.5) The v., ,, and W., i, are related to each other by ik|b ik|b ' 9''Viklb = 9'^j„b . (3.7) This allows the V.. i, to be expressed as a W., ■, In (3.1) 9ik.b-^bk|i " Ak^lb|p=°- (3.8) b Now, if the W ■ are found, then the H are known as a consequence of (3.6) r'bk^^Vn • (3-9) Equation (3.8) implies another expression for W.. i. «bk|i = 9jk.b-9^'\kW|b|k' • (3-'0) which is reinserted in place of the last term in (3.8) 9|k.b ■ ^bk|i - ^^'\k\'bri '^ ^^\k^^\b\'\\b' = ° - (3'1^> If this procedure is repeated once more, k'a ^ k'a b'p 9|k.b-9 gak%'b.i*9 93^9 Qpt^p-j^k . b'3 k'p i 'c ., /-, .„x = "bkii + 9 g^^g gp^g gci^b'k'M' • (3-'2) 23 Einstein now defines k'a . i 'a k'p >, ,_» '^iklb-Sbk.i -9 gak%'i.b^9 9359 9pkgi,b.k' ^3.13) and I'k'b' -.i' f-k' ^b' ^ i 'a k'p b'c ,, ,.x so that "'''''' ikb^i'k'Ib- =^ik|b • ^3-'5) The problem is to find U, the inverse of U, so that «ikib = ii''''''ikb^'k•|b ' ^3.16) and the knowledge of W., 1, then gives the | ., by (3.9)- Einstein's prescription for finding the inverse is presented in four pages and the answer becomes much too complicated to write down explicitly here. To quote Einstein in this paper, "Such a solution can indeed by arrived at. . .but it is cumbersome, and not of any practical utility for solving the differential equations." This statement also applies to the solutions obtained by Hishra [22] and Hlavaty[lO]. So, for all practical purposes, we are still left with the problem of sixty-four equations in sixty-four unknowns. One additional reduction is possible. The number of equations and unknowns can be reduced to twenty-four if we treat synwietric and antisymmetric components of | ., as separate quantities. [21 ] (This Is a natural thing to do since they transform separately. In fact, it can be seen from (2.10) that the anti- synwnetric components transform like a tensor, whereas the symmetric ones do not.) The fundamental tensor is also written as the sum of its parts. Consider the two equivalent equations 24 s s s s + 9isrbk^9j3rb^ + g.3rbk^9.3rbk — — — V V V V s s s <; 9kL.b-'5ki.b=93j_r kb^^siP kb^^siP kb^s^.p (3.1^) kb V ' — — ' V V ■ — V V ^ksP'bl* ^ksP bi -^ ^sP'bi* "ksP'bi ^2 -'7^ V V The sum and difference are two new equations s s s „s 9ik.b = 9skr ib*9.3Pbk + 9skr Fb^Sj^P bk (3.18) Sik.b = 9sk P ib * ^isP bk * ^skP ib * 5;, r bk (3.19) This procedure is repeated twice by permuting the indices i, k, and b. Two more pairs of equations are obtained. s s s s %b.l = hb r ki ^ \s P lb ^ ^.bP ki * \s P ib (3.20) — — — — — V V V V ^kb.l = ^sbP'kl * 9ks P'ib * ^sbp'kl ^ ^sP'ib (3-20 and > :> 3 9 9bl,k=9sirbk*^bsPki^93jP bk^gtsP ki (3-22) V V V -V Sbi.k " ^si r bk ■" hsY ki ■" 93. f bk -^ ^sP ki (3.23) The following combinations are taken s s s "^ik.b * ^kb.i ^ ^bi.k = 2[g3bP ik * SsiP bk ^ ^ksP ib^ . (3.24) — — V V V V J 25 Using (2.62) and (2.6l) as definition, Substitution of (3.25) into (3.19) and use of definition (2.9) for a covariant derivative gives the set of twenty-four equations, P P f" 9ik;b = 9^r ib ^ ^i^r bi< "■ 9^ ^W^mb P is m mm V V V ^ ^ V V b to he solved for tiie twenty-four ., . When they are found, substi- ^ b tution in (3-25) yields P . . Therefore the sixty-four r . are known once (3.26) is solved. This still does not give a tractable form for solving (3-2). Einstein's hypothesis cannot be tested unless we find some way of solving the differential equations and these in b turn cannot be solved until a useful, general form for the J ., is ob- tained. We are not completely stymied by the formidabi 1 ity of the equations. A further advancement can be made. It can be shown that, within the frame- work of the theory, the gravitational and electromagnetic fields are con- tained in the single tensor, g., , and the field equations are those of electrodynamics as well as gravitation. To see how this comes about, an alteration in the fundamental tensor is made. It is chosen to be a CI ifford-Hermitian tensor. This means the ant isytrmetric part is chosen as ^ik^^ik' <^-27) V where e is so small that its squares and higher powers will be neglected. Then the tensor, 26 9fk=9j_, + ^;k . (3.28) is called a Cl ifford-HermJtian tensor after W. K. Clifford. For the Cnfford-Hermit ian field, (3-26) becomes ^^k;b = Vi^ib^'lEP^k • <3.29) V V If i, k, and b are permuted, two more equations are obtained. ^*bKk - vr'br vr\i "•"' The following combination of the three equations is taken i^^*ik;b-*kb;i "\i;k>'3Hbr'ik- ^^'^'^ V Which is immediately solved for b r ik=2^9^(*ik;p -*kp;i -%i;k) • (3-33) This result causes (3.25) to become \ \k (i k) • (3.34) and the displacement field is known r'ik' V^^^'^'iXip'V;! -*Pi;k> • (3-« This is a general form insofar as no particular coordinate system has been specified. It is also in a useful form for the field equa- tions (3.2). (3.3). and (3.4). .-J^' 27 First consider (3.2). If (3-^) is kept in mind, then R., is given by R., = r (Rj, + R, .) ik 2 ^ jk ki '"ij'm ■'•*'""' (3.36) If i or k take on the value "k", the second part of (3-2) is identically satisfied. Equation (3-36) is a second order partial differential equation in the g., . It is easily verified that |.Pp^= logfAgJj . (3.37) so that In analogy with general relativity, g fnust be interpreted as the metric IK in the space. If we choose the physical space to possess spherical s^nmetry and the time dimension to be uncurved then the line element o may be brought into the form -ds^ = (^^(S^ + (K sin^de^ + (^ sin^sin^ed*^ - c^dt^ (3-39) in which case the metric is 8. "(5^" is interpreted as the radius of this world. 2d 'jk + (C +(R?sin^f +afsin^sin^9 -1 (3.^) The left hand side of (3-38) is calculated using (3.^0) and indeed the differential equations are satisfied by (3-40). The form (3-39) repre- sents what is called an Einstein cylindrical model of the universe. The contravariant tensor to (3-^0) is simply ?k (3.41) ik since g., is represented by a diagonal matrix, (3-^0). With the g.,^ solved, we can move on to (3.4), using (3.33) 1 pk k r ., = = jeg^ (<f., - 0, . - . ,) Itk 2 ^ ^ik;p kp;i pi;k' V (3.42) 3^—- I Since g^— is symmetric and 4>. is antisyrmietric, the second term Kp vanishes. Furthermore, p and k are dumny indices so that the first and last terms add. 1 ik;p (3.43) In this form the field equation is significant since it is recognized as one of the covariant electrodynamic field equations in the absence of charges and currents [19], with the 0., interpreted as components t K of the electromagnetic tensor 9^. ik: = (3.44) 9. The solution will be examined in more detail in the next chapter. 29 The remaining set of field equations (3-3) may be written '^i^k;fa^\b;i "*-\i;l<=-^<*lk;b-^\b:i **bi;k) " (3-^5) From definition (2.21), and (3.4) and the results of the Cllfford- Hermitian field, it follows ,P rjPCrv") nP R 4(il'p],k-lk'pli) (3.46) The term in parentheses vanishes due to (3-38) and the remaining term in brackets wi 1 1 be recognized as the definition of the covariant p derivative of the tensor, \ . ik V V V Then by (3.45) ■^ik" T ik;p (3-^7) P p p ^P l^k-.pb-^r kb;pi -^r bi;pk> ==|T<^•k;b*\b;l ^^bijk^ V (3.48) It remains to be shown in the next chapter that a solution of (3.47) is the second set of covariant electrodynamic equations. So far, by use of a CI if ford-Hermit Ian tensor, we have been able b to present a reasonable form for the P ., and a solution to field I ik ^ equation (3.2). Also (3-3) and (3.4) have been put Into a more familiar form and are ready to be solved in the next chapter by use of a special coordinate system which will make the equations especially transparent. ) CHAPTER k SPECIAL SOLUTIONS In the preceding chapter the field equations were fashioned into a suitable form for solution. The three equations which must now be solved are -KlpMi^UMr^ft\i.W^,\i=-^=H,. 0.38) \ <^ k ^ k J r" .pp Minkowski Space First, a very special situation will be considered. It is the opposite of the limiting case where the antisymmetric field vanishes. Now, the field equations will be examined when "(^' Js chosen Infinitely large and the Cl i f ford-Hermit i an tensor Is assumed to have Minkowski form, 9ik^^k-^^*lk • ^^■''> In light of this, all Christoffel symbols vanish and (3.38) is satis- fied Identically. There is no gravitational field. It Is to be expected, since there is no gravitational field, that what is left of the tensor g.. should represent the electromagnetic 30 i 31 field, and the field equations should be Maxwell's equations In a Minkowski space. To see that this Is so, let us Investigate {3.kk) and (3.48) for 10 ^Ik^^k (^•2) First, It Is seen that there Is no distinction between contravarlant and covarlant tensors. Furthermore, any covariant derivative may be replaced by an ordinary derivative since all Christoffel symbols are zero. Equation (3.M+) can be expressed J ) ^, = (^•3) Indeed, If the ^^^ represents the electromagnetic field, *lk- -le, -le. ie, le. Ie, • e. (4.4) 1 -2 -3 then the second set of field equations (3.44) becomes one set of the Maxwell's equations in an empty space, as expected c Vx h = r g ; for i = 1, 2, 3 V " e = for I a= 4 (^.5) (4.6) The third field equation, (3.48) can be written 10. We wl 1 1 choose x = let ) 32 <r',k.b^rVi*r\i,k),p^^c,k.b*Vi**bi,^> ■ <'•'' The solutions, (3.33) for P j^ are substituted. usJng 6' for g— In accord with (4.2). f'(*lk.S - *k5.I - ♦sl.k'.b * (♦kb.S - ♦bS.k - *sk,b),l * <*b>.s - *ls.b - ♦sb.i'.kl.s * ^ <*lk,b * \b.i * *bl,k' ("■»> or 2 ^ lk,b * *kb,i bl,k\ss ^ (5^2 ^ ik.b kb,i bi.k' A solution to this equation Is Ik.b kb.i bl.k which, In view of {k.k) , represents the other set of Maxwell equations In a Minkowski space. V • 1^= ; I, j, k 5^ ^ (^-11) Vx?= .i|j ; i. j. or k=4 (4.12) c ^ Therefore, as expected, the field equations in Minkowski space become the Maxwell equations, while the fundamental tensor represents ^ both the metric and the components of the electromagnetic field. These equations, (4.5), (4.6), (4.1!) and (4.12) are well known — their solution and validity need no elaboration. } 13 A More General Case We now move to a more general case, where "^" is taken as finite. A solution to (3.38) for the q^^ has been given in chapter three, but a slightly different version of the solution will be given here. First, note must be tal<en that (3-38) Involves only the symmetric part of the fundamental tensor. By an extension of the example in Minkowski space, we should anticipate that the gj,^ will depend upon the distribution of matter In the universe. At this point the following model is adopted-- a static homogeneous universe. This means that all parts are considered extrinsical ly and permanently alike. In this case, the line element can be put In the general spherically synmetric form. -ds^ = e^dr^ + r^de^ + r^sin^0dcf^ - eVdt^ {k.n) where V = v(r) and \ = X(r) . (^-l^) Most of the g.. are already known: g = , I 5^ k (^-15) 2 q«« = r (^.16) J 933 = r^sln^e . (^-17) From the differential equations, (3-39), we need only determine and Si^'e"- ^^'^^^ ) ) 3k The Chrlstoffel symbols are computed according to (2,6l) and substituted in (3-38). It is seen that the solution is 2 \ f. r v-1 e =('--52) V e = Thus in matrix form, the g., are ih. 2 (4.20) (4.21) 'Ik 2 . 2^ - sin o -1 (4.22) This model is known as the Einstein cylindrical universe. (See [24], page 359) A change of variables will serve to show more distinctly the character of this universe. Let x, = r cos 6 x_ = r sin© cos* X. e r s i n 6 s i n4> (4.23) In which case, -ds = (dx. + dx- + dx, + dx. ) - c dt (4.24) 35 and Xj + Xg + x^ + x^ = g^ . (^.25) Thfs illustrates that the physical space of the Einstein universe may be interpreted as the three-dimensional bounding manifold of a sphere % of radius (j\. in a four-dimensional Euclidean space with the cartesian coordinates given above. The time dimension is uncurved. Hence the name cylindrical universe--curved space and straight time. Even though the line element displays spherical symmetry, there equations more transparent, we adopt a method used by Kronsbein 02]. 12 The sphere represented by (4,25) is radially projected from its center in the four-dimensional space, (x.), into the three-dimensional space, \ €, with coordinates X , (Greek letters take on values 1, 2, 3) by the projection Is no symmetry of form among the ^. |^\ and the R., . To make the field The solution shown In the previous chapter, (3.4l)f may be ob- tained directly from {h.lk) by the transformation Xj = ^cos f x^ = (Jt s I n^ cos X- = <Jlsin^sin0cos¥ ^ \% = 5lsin^sln0 sinY which expresses the spatial part in four-dimensional spherical coordi nates. 12 Some people prefer to call this a gnomonic projection. 36 ^^ (R. A 1/2 (4.26) where A = (S(^ + (xV+ (X^)^ + (X^)^ (4.27) In this space, X^ and X = ct , we wi 1 1 denote the symmetrFc part of the fundamental tensor by a., , It Is computed by the standard method to give J Ik A [A- (x')2] x'x^ x'x^ XX [A- (X^)^] .x^x3 2 3 •X X^ xV [A- (X^)^] (4.28) The determinant Is i^kf = -4 (4.29) The contravariant tensor to (4.28) Is found to be '} Ik A [(R? ■»• (x')2] x'x^ x'x3 x'x^ [(Si} + (x^)^] xV x'x^ xV [(R + (X^)^l 9t A (4.30) 37 , With (4.28) and (4.30), the Chrlstoffel symbols can be constructed. They are very important because they are the symmetric part of the 13 displacement field. 21. [a ay (4.31) There is one more quantity which we need to calculate in this space for future use. It is the analog of the Riemann-Chr istoffel tensor, (2.20). It will be denoted by ll<b " jt^J.b " jl^bj.k - p 4^4" I'^^^t^^'') (4.32) Notice the antisymmetry. Ikb ibk (4.33) i The components will be listed here for future reference: a B XX- aBT ' .2 oaT a2 Al 1 others are zero (4.34) For the antisymmetric part of the fundamental tensor, the following symbols will be used for the '^ ik' 1 ■? In (4.31) and (4.34), the repeated index does not imply summation. 38 -h. 3 . (^.35) where the "e^^" and "h^" are functions of (x\ X^, X^ , t). We now have the necessary material to calculate the antisymmetric b components of | ., in our special coordinates. The actual calculation of all twenty-four P ,, is a long, tedious process. Only a sample 'v 3 calculation (the component, | .j' ^' ^ ^ ^^ ^^^"^ ^^ =•" example) and the final results are given here. By (3-33), r 12= ^ta3'.,^^, .a32.,^.^^.la33(.^^^, .*,,.,..„. J] . (4.36) ;3 23;1 31;2' r,2-^^^'^^2;l* A2;2*^'^ti(<^12,3"'23J -*31.2) *i2 {13)" II \3 23 ]} . (^.37) Upon substitution of (4.30), (4.31), and (4.35), and consequent can- cellation and simplification, It is found that - le} + (X^)^][y(h,^, + h2^2 + SJ * S ^^^^] + ^\ +x3h^)]) . (4.38) b The other ., are computed In the same manner and soon the following \ ) pattern Is recognized 39 r e, = ^[^^(3X\ . a"\_,) - c^a<*(i V,. - i «V 1 (^-33) and -'' rA .A X^ aM- 1 oM- i ^"^a^h^^^] {h.kO) also P V and These are the twenty-four jl- '" ^^^ special coordinates. Combined b I K V with the n .. given by (^.30. the complete set of the sixty-four is known in this space. This completes the information needed to set up the field equations, (3-^) and (3.^8). It has been pointed out that (3.^). which reduces to (3-^5) Is identically a set of the covariant electrodynamic equations in the presence of a gravitational field. To see what they look like in this space, It will be easier to go back to (3.^), since the | ., have ik already been computed. The equations represented are: r'i2*r'i3^rV=° <"■'"> '*^\ and € are the usual permutation symbols whose value is zero if any two indices are alike, +1 if (<^BA.) is an even permutation of (123), -1 If (°=BX) is an odd permutation of (123). 40 } \ y ?\^^V\,-V\k-^ i'-^) r\, -r'32-^r 34=° ^^-^5) 31 I 32 ^1 34 4] + I 42 -I 43 r\, -r\2±r\3 = ° ^^•'^^^ ,b The n ,, are substituted from equations (4.39) to (4.42). After simplification, the four equations are: 3(X^h3 - X^^) + f®-^ + (X^)^^'^3.2 " ^^^ ■" (X^)^^h2^3 ^xV(h3 3 -h^^) ^X'(X% , -X^h^.,) -^e,^^=0 (4.47) 3(X^h, - x'h^) + [(SL^ + (X^)^]h, 3 - [(R.^ + (xV]h3 , . . x3x'(h, , - h3 3) . x2(x^,_2 - >^'^^3,2) - 4^ -2.4 = ° (^-^S) 3(X^2 - X^h,) + l^^ + (xV] h2 , - [(«.^ + (X^)^]h, 2 *x'x^h2^2 -^j) •^X3(x'h2^3 -X^h, 3) .-^33^^=0 (4.49) 2(xU, ^ X^e^ + X^e3) + [(SL^ + (xV e, , + x'x^e, 2 + X^X^e, 3 + X^x'e2 , + [(Sl^ + (X^)^]e2^2 + ^^^^®2,3 + X^x'e3 , + X^X%3 2 + IS{} + (x^^]e3 3 = 0. (4.50) These equations are exceedingly complicated as they stand. They will be left this way for the time being. After the other set of four field equations (3.48), has been written in this space, a simplification and solution will be presented for all eight equations. A tensor identity which will be of great utility in the investi- gation of (3.48) is ^1 .Ik--- .Ik--. Ik--- ^s ik--- ^- A . -A . =A G,+A 1.GS + bnT--;pq bm- • • ;qp sm- ■ - bpq bs--- mpq A^k- - • _ ! „ is- - - „k /I ^,x - A , G - A , G - • - • (4.5 ) bm- • spq bm- - - spq ^ "^ ' i k- • • s where A . is any tensor of arbitrary rank and G is defined bm- • • ' tpq by (4.32). This identity allows the left hand side of (3.48) to be written I ik;pb*l kb;pi "^l bl;pk°M ik;b"^' kbii"^' bi;k^p V V* V s^ V •• *l tk*^ Ipb *l it° kpb *' U>°''tbp • tk kpi ' kt bpl ' kb tip ^ If g'^j. Is written out as prescribed In (4.32), then it Is seen to be '^'ikp-^k (^-53) It Is clear that If any of the i, k, b are equal, then (4.52) vanishes, as does the right hand side of (3.48). Equation (3.48) represents only four distinct equations. They are: r'lj;p2-r'3j;pl ^^'2J;p3-|I(^3;2^^2;l*^l;3) ^''''^ y r'l^;p2 ^ r\2;pl •^r'21;p/, = f2 (^4,2■^^2;l * *21 54^ ^^'^S) r 43;p2 *P 32;p4 ^P 24;p3 = % ^*43;2 * ^32;4 + *24;3^ ^""'^^^ The left hand side of (4.54) is expanded like (4.52). ^ Hi p I 13;p2'^n 3J;pl ■^n 2J;p3"M 1^;2"^| 3J;1 ^l 21 ;3^ 1 . ^3 . ^ 1 . t-r 3 *n ,^^'uz^V t3^'l32*r 1/312 *P 1/332 >^ V V V , ,1 n 2 r73 + 1 ' ..G^„, -hI ' ..G\„ +1 ' ,.G\., +1 ^.. 231 I 13*^12 *l 13^22 * U"^!?. "'V321*r\2^'331 *r 3t<^'221 ^P 3/ 1 2 3 "■I 32^1 *n 32'^21 *l 32^^31 *r"t,s'2,3 -r't,='2„ *r '2/113 ^r \/ 2 r-i3 +r zi^^n^n 2i'^23*r 3i'^33 ^^^-^^^ ^13 ' 21 "23 I 31 33 .s If the index "4" appears In a G ^ , it vanishes. For this reason, the tpq following type of combinations must be observed. g',2, .g3,,3 = R^ (4.59) • When the summation is carried out on the index "t" as Indicated In (4.58), and combinations like (4.59) heeded, a fortunate cancellation occurs which leaves r',3-,p2 ^r'32,p, *r'2..p3 = f'^.a f '32;, ^r'2U3';p (^■'°' V V V V V V ^ The field equation (4.54) Is then <r'>3.2*r'32;, *r'2.;3>;p-&<*.3;2**32;, **2,,3'- (^•^" b At this point, the set of M ., , as given In equations (4.39) V through (4.42), could be substituted, but the left hand side of (4.61) i J i»3 can still be reduced. By (3.33), p r- 1 s^Pr* +4) +d) -d> -* [I3;2];p 2^.M3;52* 32;sl '^21;s3 3s;12 *sl;32 The summation Is carried out, as indicated, over s. Tlien (4.51) is used to make tiie following type of rearrangement ab;cd ab;dc pb acd ap bed throughout each term in (4.62), giving r'[l3;2];p = f ^''^B^ahsp * ^'="'(*13='n2 * V'lU * *I2='312 * ^1°^13 * ''21^^l3 -^ V^13^P " ^''^'l2«'321 * *3iG'221 * V'221 ^ ^1^'223 ^ *31^^23 * V^23^P * a3^*23^'l32 * *12'^'332 * ^3'^^32 "^ ^2<^^31 *v'33l^^l^^3l^p• ^'-'^^ The covarlant derivative of the G .., are all zero, since, according to J kb equations (4.34) and (4.28) J ^ J 1 * is a constant and = «B7--:^=aB ■ C-^S) a^.p = , (4.653) 1 5 No summation Is Implied by repetition of the index 7 In these equa- tions, therefore The following equations are now substituted in {k.Gk): (4.65), (4.28), and (4.30), Cancellation and simplification leave 'P € sp^ .2 .^:2];p " 2 ^ '^[I3:2];sp "" ' This result reinserted in the left hand side of (4.54) gives finally r'[13;a].,p-f^''*[,3.2].sp*i2^n3,2] • ''•"' aSP((j) + 4) + 4) ) =0 . (4.68) ^ 13;2 32;1 21;3 ;sp The same procedure and the same Identities are used on equations (4.55), (4.56). and (4.57), giving ^''(^4;2*%2;l^^l;4>;sp = ° ^'-''^ ^''^\3;2*S2;4**24;3^;sp"° ^''^'^ ^ 13;4 34;1 4l;3 ,sp Solutions to (4.67) through (4.70) are, respectively, ■ *.3,2**32., **21,3 = ° <"•''' *24.1*%, 2**12,4-° . C-^' J V.2 * \2.3 * *23.4 = " <"■*' *I4,3*\3.1**31,4=° • ^"-^^ Using e^ and h^ from (4.35) we see that this is the second set of covariant electrodynamic equations. 45 * "2,2 * "3,3 - ° '"■''' =3.2-^2,3*"l,^=° ''••^' ; ^1,3 -^3,1* "2.4=° f^-^' These four equations are simple compared with the other set, (^.47) through (4.50). In order to simplify the latter, we are willing to slightly complicate the former. The result will be a symmetry of form for both sets, and moreover, a solution will follow easily. This is accomplished by replacing the e and h by the following quantities J e^(x';x^x^t) = -^ [«.£„ + e^iJY^e^'^^ • ^^'^^^ where H^ = H^(t) (4.82) and E, = E^(t) (4.83) are yet to be determined. These values are substituted in the field equations in place of e and h . In the first set, (4.50) is satisfied identically by (4.81) Equation (4.47), after simplification, reduces to 2H, y3 2H5 y2 2H. or 1 ht c5?^2 1 St ich implies 1 ^E* 2 -> c ^t - fit " . (^.85) (4.86) ^ This same result Is Implied by (4.48) and (4.49). In the second set of equations, (4.76) is satisfied identically by (4.80). When (4.80) and (4.81) are substituted in (4.77), the result Is [(%} + (X^)^](2E3 +<K»3,4) + ('^'x^ +<50(^)(2E, +(SLH,^^) + (xV - <8.X^(2E2 + (^H^^^^) = . (4.87) 16L' . (x3)2](2r. ^ 1^)3 . (x\^ .61 /mt. f f), + (xV -(ax')(2?+^|H')^ = , (4.88) which Impl les This result Is also Implied by (4.78) and (4.79). J These two sets of equations are written together to emphasize the symmetry of form which has been brought about. In place of the original sets, (4.47) through (4.50) and (4.76) through (4.79), we have ^7 ^ — ^ Substitution of these equations Into one another allows E and H to be eliminated from each equation respectively, giving two ordinary wave equations, 7&'W^'' (MO) ^^+-^r=o . (^.91) c^ dt^ oJ^ These represent an electromagnetic wave of circular frequency '^=.a. 2c (4.92) where E^(t) = G° sin wt (^-93) H^(t) = G^ cos wt . (^-9^) The Gff ^s a constant. So we finally have the solution to the two sets of equations, (4.47) through (4.50) and (4.76) through (4.79). it is, in the special coordinates, and With the *,. given by (4.35), the above is the solution to the last two unified field equations, (3.44) and (3-48), which we had set out to solve at the beginning of this chapter. In vector notation, the solutions are 48 h^-% [aV+ (X*. m* (h'^x^] (^-97) a2 e^^ [€^+ it^^lt)] . (4.98) The reason for using vector notation will soon be apparent. Now that the solutions are known, we want to visualize them in the spherical space. First, consider the "straight lines" in this space. The geodesic equation, d^X* . l-l' dx"" dX*" 6X' + rkbfrjr-" ■ C-s^) becomes d^x' ^f f ) dxll dX^^ (k \00) s I nee r 'kbfrfr'O ("■"") and r kb |k b\ • (3.35) This shows that the geodesies of a space are not altered by the presence of electromagnetic fields In the space to the first order which we have 1 Investigated. (A question which immediately comes to mind is, "Well, do the properties of the space affect the electromagnetic fields?" Of course they do. We have seen how the electrodynamic equations were complicated. Exactly how they are affected remains to be seen below.) From [12], the above equati'on (4.100) can be writt en ) A" 2 dX^ .-♦ dx! ^^2 " A dA ^^ • d\^ 2 k ^^= (4.102) An explicit;' Integral of (4.102) Is j,l ^ sin M- sin (X H- 6) cos (X + €) X^ B sin M- sin (X + 5) cos (X + e) X = tan (X + €) . (4.103) We liave set (J^ = 1 for simplicity. This is the mathematical form for 16 the "straight lines." A geometrical picture may be found in [l2]. Now when the sphere, (4.25) is rotated with angular velocity (^, In E^^, it Induces transformations on X in €_. Points move with el 1 ipt Ic velocities (R.(^ ■ (S^a?+ (X*. (j5x*+ <5^(i?x X^ , (4.104) It can been seen that the geodesic equations fulfill (4.104) so that the points move along straight lines In the space. Now, replace w^ln (4.104) by H*and the result Is the same as (4.97). We see that the magnetic vector, h*, has the form of the associated contravarlant elliptic velocity, (4.104), and lies along the geodesies of the space. So not only Is the magnetic vector altered by the space, It Is altered In this very special way. 16 The geodesies will sometimes be referred to as Clifford lines. ) 50 If we tensor multiply the contravar lant velocity by g, , we get the covariant velocity vector, which turns out to be the same form as the electric vector, (4.98) X. = -s^ [<SLWj + (w'x X^.] . (it. 105) so that the electric vector lies along the covariant elliptic velocity vector. The solutions for the unified field equations are, therefore, standing waves lying along Clifford lines, and having fixed frequency, 2c It may be pointed out, finally, that when the radius of the Einstein universe becomes infinitely large, the field equations and their solutions naturally go over to Maxwell's equations and their solutions in a Minkowski space. ) ; > CHAPTER 5 SUMMARY There is a possibility that the main features of the theory pre- sented have been obscured by the preponderance of raathamatics. For this reason, It may be well to summarize briefly the results of this dissertation. The second chapter contains no new results. It is intended as a background to familiarize the reader with the different versions of the unified field theory. The similarities and differences are pointed out. All three have the same goal--to derive all of the fields and field laws governing Nature from a single tensor, which need not have symmetry. The main difference is the choice of integrand for the variational principle. This, of course, leads to different forms of the field equations; however, the same set of sixty-four alge- braic equations appears in each version. The most Important similarity in the theories is that they are based on the same geometrical con- cepts, and they all go over into the gravitational equations of general relativity In a limiting case. Notwithstanding all the beauti- ful mathematical formalism, this correspondence to the laws of gravi- tation is the only physical content which has been derived from any of the theories up to now. The reason for this difficulty was pointed out in the next chapter. Before Einstein's hypothesis (which was discussed in the Introduction) can be tested, the differential field equations must be solved. Before 51 _; 52 they can be solved, the sixty-four algebraic equations must be solved in a general case (without reference to any coordinates) for the components of the displacement field. Since this general solution has not been accomplished, there is no way of telling whether all the field laws are Included in the theories. The sixty-four equations are shown to be reducible to twenty-four, but this has not enabled the solu- tion to be found up to the present. At this point, we present our version of the theory, in which the fundamental tensor is modified. It must be noted that this is not a linearization such as that used by Kursunoglu [\k] or Einstein [3]. These linearizations exclude interaction between the symmetric and antisymmetric components of g., . The present work does not preclude this possibility. In fact, the antisymmetric components of the dis- placement field were seen to be a combination of the symmetric and anti- symmetric parts of the fundamental tensor. So instead of a linearization, we have a perturbation type of technique. It is basically a first order antisymmetric perturbation of the gravitational field producing symmetric tensor. With this method, the algebraic equations are readily solved for j ., without the specification of any coordinate system. It is found that the R., become the same expressions used in general relativity. With this in mind, we adapt the set of field equations involving the contracted curvature tensor to an Einstein model of the universe. In the limit of an infinite radius of the Einstein universe, the remaining differential equations go directly over to the Maxwell equa- tions of Minkowski space. For the general case (a finite radius), a transformation was made to a symmetrical arrangement of coordinates. Investigation of the remaining two differential field laws leads to the ; 53 important discovery that tliey are tlie covariant electrodynamic equations in tlie absence of charges and currents. As a consequence of the synmetry of the coordinates, we are able to give an exact solution to these equations. This solution indicates that the electromagnetic is bent along the geodesic caused by the gravi- tational field, while an investigation of the geodesic equations shows that the gravitational field is unaffected by the electromagnetic field in this case^ Perturbation to higher order terms in € still makes it possible to obtain the displacement field explicitly but this does not, up to the present, Imply success in solving the associated differential equations. In these cases, the presence of addition fields will distort the geodesies of the pure gravitational field, but it is not known whether the additional fields are only electromagnetic in Nature. In summary, the most important result of this disseration is the realization that, to the first order, the covariant electrodynamic field equations, as well as the gravitational equations, are Included in the unified field theory. Heretofore this was conjectured but never shown. It is clear that what has been done is far from an ultimate goal of the theory. Nevertheless, we feel that our contributions should be an impetus to further work in the field. The most general case must be pur- sued. Along these lines, Kursunoglu has made the greatest Innovations since the theory was formulated by Einstein. It would certainly be desirable to study his plan in which there is a possibility of deriving nuclear fields together with those previously discussed. [16] BIBLIOGRAPHY 1. Einstein, A. The principle of Relativity . New York: Dover Publi- cations, 1923, pp. i 11-164. ^ 2. Einstein, A., Ann. Math. 46. 578 (1946). 3. Einstein, A. and Strauss. E. , Ann. Math. 47. 73' (1946). 4. Einstein, A. and Kaufman, B., Ann. Math. 59, 230 (1954). 5. Einstein, A. and Kaufman, B. , Ann. Math. 62, 128 (1955). 6. Einstein, A., Can. J. Math. 2, 120 (1950). 7. Einstein, A. The Meaning of Relativity . 5th ed. Princeton: Prince- ton University Press, 1955- a. Einstein, A., Revs. Modern Phys. iO, 35 (1948). 9. Eddington, A. S. The Mathematical Theory of Relativity . Cambridge: ^ Cambridge University Press, 1924. 10. Hlavaty, V. Geometry of Einstein's Unified Field Theory . Groningen; P. Noordhoff Ltd. , 1957- 11. Hlavaty, v., J. of Math, and Mech. 7. 833 (1958). 12. Kronsbein, J., Phys. Rev, JhOg, I8I5 (1958). 13. Kronsbein, J. Phys. Rev. J_12^, 1384 (1958). 14. Kursunoglu, B. , Phys. Rev. 88, I369 (1952). 15. Kursunoglu, B. , Revs. Modern Phys. 29, 412 (1957). 16. Kursunoglu, B., |1 Nuovo Cimento 21, 729 (I96O). 17- Papapetrou, A., Proc. Roy. Irish Ac. 52, A, 69 (1948). 18. Schrd'edlnger, E. Space-Time Structure . Cambridge: Cambridge Uni- versity Press, i960. 19. Synge, J. L. and Schlld, A. Tensor Calculus . Toronto: University of Toronto Press, 1949. 20. Tolman, R. C. Relativity, Thermodynamics, and Cosmology . Oxford: I Oxford University Press, 1934. 54 ; 55 21. Tonnelat, M. A. La Theorie Du Champ Unifie' D' Einstein . Parts: Gauthler-VMlars, 1955. 22. Mishra, R. S., J. of Math, and Mech. 7, 877 (1958). 23. Landau, L. D. and Lifshltz, E. M. The Classical Theory of Fields , 2nd ed. Reading, Mass.: Add i son-Wesley Publishing Company, Inc., 1962. 2k. M«5ller, C. The Theory of Relativity . Oxford: Oxford University Press, 1952. 25. Kronsbein, J. Electromagnetic Fields in Einstein's Universe [Unpubl ished]. ; BIOGRAPHICAL SKETCH Joseph Francis Pizzo, Jr. was born on October 30, 1939, In Houston, Texas. There he attended St. Thomas High School and received his B. A. from the University of St. Thomas in I96I. In September, I96I , Mr. Pizzo began graduate studies at the University of Florida. He received his Ph.D. in August, 1964. Mr. Pizzo Is married to the former Paula Awtry of Dallas, Texas. They have one child, a son. 56 This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Phi losophy. August 8, 1964 Dean, Col leg^'Xbf ,.A'rt& and Sciences Dean, Graduate School Supervisory Cormittee: Cha irman L^ FW^ Co-chairma ^ f^.S.T Cm-'-Av-v ' liv ei.v_.v C vU F '//J^c^/l.A